ARITHMETIC PROGRESSIONS IN SUMSETS AND DIFFERENCE SETS

Abstract

Let s ≥ 1, Ai ⊆ {1,2,…,N} (1 ≤ i ≤ s) and k ≥ 3 be an odd integer. In this paper, one of the main results is: if [Formula: see text], then (a) each of Ai - Aj (1 ≤ i, j ≤ s) contains an arithmetic progression of length k with the same common difference; (b) all sets Ai - Ai (1 ≤ i ≤ s) contain a common arithmetic progression of length k. Finally, we pose several problems for further research.